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## Scholarly Works (11 results)

The monopole-dimer model is a signed variant of the monomer-dimer model which has determinantal structure. We extend the monopole-dimer model for planar graphs (Math. Phys. Anal. Geom., 2015) to Cartesian products thereof and show that the partition function of this model can be expressed as a determinant of a generalised signed adjacency matrix. We then show that the partition function is independent of the orientations of the planar graphs so long as the orientations are Pfaffian. When these planar graphs are bipartite, we show that the computation of the partition function becomes especially simple. We then give an explicit product formula for the partition function of three-dimensional grid graphs a la Kasteleyn and Temperley-Fischer, which turns out to be fourth power of a polynomial when all grid lengths are even. Finally, we generalise this product formula to \(d\) dimensions, again obtaining an explicit product formula. We conclude with a discussion on asymptotic formulas for the free energy and monopole densities.

Mathematics Subject Classifications: 82B20, 05A15, 05C70

Keywords: Monopole-dimer model, cartesian products, determinantal formula, Kasteleyn orientation, bipartite, cycle decomposition, partition function, grid graphs, free energy

- 1 supplemental ZIP

Recall that an excedance of a permutation \(\pi\) is any position \(i\) such that \(\pi_i > i\). Inspired by the work of Hopkins, McConville and Propp (Elec. J. Comb., 2017) on sorting using toppling, we say that a permutation is toppleable if it gets sorted by a certain sequence of toppling moves. One of our main results is that the number of toppleable permutations on \(n\) letters is the same as those for which excedances happen exactly at \(\{1,\dots, \lfloor (n-1)/2 \rfloor\}\). Additionally, we show that the above is also the number of acyclic orientations with unique sink (AUSOs) of the complete bipartite graph \(K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}\). We also give a formula for the number of AUSOs of complete multipartite graphs. We conclude with observations on an extremal question of Cameron et al. concerning maximizers of (the number of) acyclic orientations, given a prescribed number of vertices and edges for the graph.

Mathematics Subject Classifications: 05A19, 05A05, 05C30

- 1 supplemental ZIP